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Approximate gravitational potentials are often used to describe analyticallythe motion of particles near black holes (BHs), as well as to study thestructure of an accretion disk. Such 'pseudo-Newtonian' potentials are usedwith the flat-metric equations. Here we consider the motion of a free particlenear a non-rotating BH in the context of an exact `logarithmic' gravitationalpotential. We show how the logarithmic potential gives an exact solution for amechanical problem and present the relativistic Bernoulli equation for thefluid in the Schwarzschild metric.

Logarithmic potential for the gravitational field of Schwarzschild black holes